Using Frobenius Reciprocity, derive the character table of S_n from that of A_n when

(a) n = 3

(b) n = 4

(c) n = 5

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- Dec 15th 2009, 03:14 PMakc2010characters of S_n from A_n
Using Frobenius Reciprocity, derive the character table of S_n from that of A_n when

(a) n = 3

(b) n = 4

(c) n = 5 - Dec 15th 2009, 10:10 PMaliceinwonderland
This is for n=5.

The character table for S_5 is

The character table for A_5 is here.

I followed the notations of the book, Fulton's "Representation Theory : A first course".

Note that the conjugacy class (12345) in S_5 with size 24 splits into two conjugacy classes (12345) and (21345) in A_5 with size 12, respectively. Note also that the conjugacy classes (12), (1234) and (12)(345) are not available in A_5.

Let H=A_5 and G=S_5. Then, by using Frobenius's reciprocity

Res U = U, Res U' = U, Ind U = U U',

Res V = V, Res V' = V, Ind V = V V',

Res W = W, Res W' = W, Ind W = W W', and

Res = Y Z, Ind Y = Ind Z = .

The last line is obtained by using the lemma,

,

where is the order of the conjugacy class in G of g,

are representatives of s conjugacy classes of H,

be the orders of these classes.